1.2.1 Tensor Definition

A tensor is a mathematical object that generalizes vectors and matrices, defined by its components and transformation rules under coordinate changes.

Tensor Definition is the precise mathematical characterization of a tensor as an element of a tensor product space built from a vector space and its dual, or equivalently, as a multilinear map on an appropriate combination of vectors and covectors that satisfies specific transformation rules under a change of basis. It fixes, in exact terms, what qualifies as a tensor, distinguishing genuine tensorial objects from indexed arrays of numbers that merely resemble them.

The Coordinate-Free Definition

The most general and abstract definition characterizes a tensor as an element of a tensor product of several copies of a vector space and its dual space. A tensor of type (p, q), sometimes called a tensor of rank p+q, is an element of the tensor product of p copies of the vector space and q copies of its dual. This definition makes no reference to any particular basis or coordinate system; the tensor exists as an abstract algebraic object, and a choice of basis is needed only to express it in terms of numerical components.

T \in \otimes_{i = 1}^{p} V \otimes \otimes_{j = 1}^{q} V^{*}

The expression above states, in symbolic form, that a tensor of type (p, q) belongs to the tensor product of p copies of the vector space and q copies of its dual space.

The Multilinear Map Definition

An equivalent and frequently used definition characterizes a tensor of type (p, q) as a multilinear map that takes p covectors and q vectors as arguments and produces a scalar. This formulation emphasizes the tensor's role as a machine that consumes the appropriate number of vectors and covectors and outputs a number in a way that is linear in every argument separately. The equivalence between this definition and the tensor-product definition follows directly from the universal property of the tensor product, which establishes a natural correspondence between multilinear maps on a collection of spaces and linear maps on their tensor product.

The Coordinate-Based Definition

A third, more elementary definition, common in physics and engineering, characterizes a tensor through its components relative to a chosen basis, together with a specified rule describing how those components must change when the basis is changed. Under this definition, an indexed collection of numbers qualifies as the components of a tensor only if it transforms according to the tensor transformation law — a rule involving the coefficients that relate the old and new bases, applied once for each index, with upper indices transforming contravariantly and lower indices transforming covariantly.

This coordinate-based definition is logically equivalent to the coordinate-free definitions, since the tensor transformation law is precisely the condition that guarantees the components describe one and the same abstract multilinear object regardless of which basis is used to express it. Its practical value lies in making tensors directly usable in explicit calculations, at the cost of temporarily obscuring the coordinate-independent nature of the underlying object.

Why the Transformation Rule Is Essential

The requirement that components transform correctly under a change of basis is what separates a tensor from an arbitrary array of numbers indexed by several subscripts. Two different bases will, in general, assign entirely different numerical components to the same underlying tensor, yet those components are related by a fixed, predictable rule determined solely by the change of basis. An array of numbers that fails to obey this rule under a change of basis does not represent a tensor, however many indices it may carry, because it does not correspond to a single, basis-independent multilinear object.

Special Cases

The general definition of a tensor recovers several familiar objects as special cases. A tensor of type (0, 0) is a scalar, invariant under every change of basis. A tensor of type (1, 0) is an ordinary vector, and a tensor of type (0, 1) is a covector. A tensor of type (1, 1) can be represented, once a basis is fixed, as a square matrix, corresponding to a linear map from the vector space to itself. Higher-rank tensors extend this pattern to multilinear relationships involving three or more vector and covector arguments, with no upper limit on the rank that can, in principle, be considered.